3.29.63 \(\int \frac {\sqrt {-b+x^2} (c+x^4) \sqrt {x+\sqrt {-b+x^2}}}{x} \, dx\) [2863]

Optimal. Leaf size=301 \[ \frac {2 \sqrt {-b+x^2} \left (-1368 b^4 x+10395 b^2 c x+3705 b^3 x^3-32340 b c x^3+1335 b^2 x^5+18480 c x^5-8100 b x^7+5040 x^9\right )+2 \left (304 b^5-2310 b^3 c-3078 b^4 x^2+24255 b^2 c x^2+3735 b^3 x^4-41580 b c x^4+4755 b^2 x^6+18480 c x^6-10620 b x^8+5040 x^{10}\right )}{3465 \left (x+\sqrt {-b+x^2}\right )^{9/2}}+\sqrt {2} b^{3/4} c \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x+\sqrt {-b+x^2}}}{-\sqrt {b}+x+\sqrt {-b+x^2}}\right )+\sqrt {2} b^{3/4} c \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {x}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt {-b+x^2}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt {x+\sqrt {-b+x^2}}}\right ) \]

[Out]

1/3465*(2*(x^2-b)^(1/2)*(5040*x^9-8100*b*x^7+1335*b^2*x^5+3705*b^3*x^3+18480*c*x^5-1368*b^4*x-32340*b*c*x^3+10
395*b^2*c*x)+10080*x^10-21240*b*x^8+9510*b^2*x^6+7470*b^3*x^4+36960*c*x^6-6156*b^4*x^2-83160*b*c*x^4+608*b^5+4
8510*b^2*c*x^2-4620*b^3*c)/(x+(x^2-b)^(1/2))^(9/2)+2^(1/2)*b^(3/4)*c*arctan(2^(1/2)*b^(1/4)*(x+(x^2-b)^(1/2))^
(1/2)/(-b^(1/2)+x+(x^2-b)^(1/2)))+2^(1/2)*b^(3/4)*c*arctanh((1/2*b^(1/4)*2^(1/2)+1/2*x*2^(1/2)/b^(1/4)+1/2*(x^
2-b)^(1/2)*2^(1/2)/b^(1/4))/(x+(x^2-b)^(1/2))^(1/2))

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Rubi [A]
time = 0.85, antiderivative size = 391, normalized size of antiderivative = 1.30, number of steps used = 18, number of rules used = 12, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {6874, 2145, 473, 470, 335, 303, 1176, 631, 210, 1179, 642, 459} \begin {gather*} \sqrt {2} b^{3/4} c \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {\sqrt {x^2-b}+x}}{\sqrt [4]{b}}\right )-\sqrt {2} b^{3/4} c \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {\sqrt {x^2-b}+x}}{\sqrt [4]{b}}+1\right )-\frac {b^{3/4} c \log \left (\sqrt {x^2-b}-\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {x^2-b}+x}+\sqrt {b}+x\right )}{\sqrt {2}}+\frac {b^{3/4} c \log \left (\sqrt {x^2-b}+\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {x^2-b}+x}+\sqrt {b}+x\right )}{\sqrt {2}}-\frac {b^5}{144 \left (\sqrt {x^2-b}+x\right )^{9/2}}-\frac {b^4}{80 \left (\sqrt {x^2-b}+x\right )^{5/2}}+\frac {b^3}{8 \sqrt {\sqrt {x^2-b}+x}}-\frac {1}{24} b^2 \left (\sqrt {x^2-b}+x\right )^{3/2}+\frac {1}{3} c \left (\sqrt {x^2-b}+x\right )^{3/2}-\frac {b c}{\sqrt {\sqrt {x^2-b}+x}}+\frac {1}{176} \left (\sqrt {x^2-b}+x\right )^{11/2}+\frac {1}{112} b \left (\sqrt {x^2-b}+x\right )^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[-b + x^2]*(c + x^4)*Sqrt[x + Sqrt[-b + x^2]])/x,x]

[Out]

-1/144*b^5/(x + Sqrt[-b + x^2])^(9/2) - b^4/(80*(x + Sqrt[-b + x^2])^(5/2)) + b^3/(8*Sqrt[x + Sqrt[-b + x^2]])
- (b*c)/Sqrt[x + Sqrt[-b + x^2]] - (b^2*(x + Sqrt[-b + x^2])^(3/2))/24 + (c*(x + Sqrt[-b + x^2])^(3/2))/3 + (
b*(x + Sqrt[-b + x^2])^(7/2))/112 + (x + Sqrt[-b + x^2])^(11/2)/176 + Sqrt[2]*b^(3/4)*c*ArcTan[1 - (Sqrt[2]*Sq
rt[x + Sqrt[-b + x^2]])/b^(1/4)] - Sqrt[2]*b^(3/4)*c*ArcTan[1 + (Sqrt[2]*Sqrt[x + Sqrt[-b + x^2]])/b^(1/4)] -
(b^(3/4)*c*Log[Sqrt[b] + x + Sqrt[-b + x^2] - Sqrt[2]*b^(1/4)*Sqrt[x + Sqrt[-b + x^2]]])/Sqrt[2] + (b^(3/4)*c*
Log[Sqrt[b] + x + Sqrt[-b + x^2] + Sqrt[2]*b^(1/4)*Sqrt[x + Sqrt[-b + x^2]]])/Sqrt[2]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 473

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[c^2*(e*x)^(m
+ 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^p*Simp[b
*c^2*n*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*(m + 1)*d^2*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && Ne
Q[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && GtQ[n, 0]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 2145

Int[(x_)^(p_.)*((g_) + (i_.)*(x_)^2)^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :>
Dist[(1/(2^(2*m + p + 1)*e^(p + 1)*f^(2*m)))*(i/c)^m, Subst[Int[x^(n - 2*m - p - 2)*((-a)*f^2 + x^2)^p*(a*f^2
+ x^2)^(2*m + 1), x], x, e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0
] && EqQ[c*g - a*i, 0] && IntegersQ[p, 2*m] && (IntegerQ[m] || GtQ[i/c, 0])

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\sqrt {-b+x^2} \left (c+x^4\right ) \sqrt {x+\sqrt {-b+x^2}}}{x} \, dx &=\int \left (\frac {c \sqrt {-b+x^2} \sqrt {x+\sqrt {-b+x^2}}}{x}+x^3 \sqrt {-b+x^2} \sqrt {x+\sqrt {-b+x^2}}\right ) \, dx\\ &=c \int \frac {\sqrt {-b+x^2} \sqrt {x+\sqrt {-b+x^2}}}{x} \, dx+\int x^3 \sqrt {-b+x^2} \sqrt {x+\sqrt {-b+x^2}} \, dx\\ &=\frac {1}{32} \text {Subst}\left (\int \frac {\left (-b+x^2\right )^2 \left (b+x^2\right )^3}{x^{11/2}} \, dx,x,x+\sqrt {-b+x^2}\right )+\frac {1}{2} c \text {Subst}\left (\int \frac {\left (-b+x^2\right )^2}{x^{3/2} \left (b+x^2\right )} \, dx,x,x+\sqrt {-b+x^2}\right )\\ &=-\frac {b c}{\sqrt {x+\sqrt {-b+x^2}}}+\frac {1}{32} \text {Subst}\left (\int \left (\frac {b^5}{x^{11/2}}+\frac {b^4}{x^{7/2}}-\frac {2 b^3}{x^{3/2}}-2 b^2 \sqrt {x}+b x^{5/2}+x^{9/2}\right ) \, dx,x,x+\sqrt {-b+x^2}\right )+\frac {c \text {Subst}\left (\int \frac {\sqrt {x} \left (-\frac {3 b^2}{2}+\frac {b x^2}{2}\right )}{b+x^2} \, dx,x,x+\sqrt {-b+x^2}\right )}{b}\\ &=-\frac {b^5}{144 \left (x+\sqrt {-b+x^2}\right )^{9/2}}-\frac {b^4}{80 \left (x+\sqrt {-b+x^2}\right )^{5/2}}+\frac {b^3}{8 \sqrt {x+\sqrt {-b+x^2}}}-\frac {b c}{\sqrt {x+\sqrt {-b+x^2}}}-\frac {1}{24} b^2 \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{3} c \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{112} b \left (x+\sqrt {-b+x^2}\right )^{7/2}+\frac {1}{176} \left (x+\sqrt {-b+x^2}\right )^{11/2}-(2 b c) \text {Subst}\left (\int \frac {\sqrt {x}}{b+x^2} \, dx,x,x+\sqrt {-b+x^2}\right )\\ &=-\frac {b^5}{144 \left (x+\sqrt {-b+x^2}\right )^{9/2}}-\frac {b^4}{80 \left (x+\sqrt {-b+x^2}\right )^{5/2}}+\frac {b^3}{8 \sqrt {x+\sqrt {-b+x^2}}}-\frac {b c}{\sqrt {x+\sqrt {-b+x^2}}}-\frac {1}{24} b^2 \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{3} c \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{112} b \left (x+\sqrt {-b+x^2}\right )^{7/2}+\frac {1}{176} \left (x+\sqrt {-b+x^2}\right )^{11/2}-(4 b c) \text {Subst}\left (\int \frac {x^2}{b+x^4} \, dx,x,\sqrt {x+\sqrt {-b+x^2}}\right )\\ &=-\frac {b^5}{144 \left (x+\sqrt {-b+x^2}\right )^{9/2}}-\frac {b^4}{80 \left (x+\sqrt {-b+x^2}\right )^{5/2}}+\frac {b^3}{8 \sqrt {x+\sqrt {-b+x^2}}}-\frac {b c}{\sqrt {x+\sqrt {-b+x^2}}}-\frac {1}{24} b^2 \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{3} c \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{112} b \left (x+\sqrt {-b+x^2}\right )^{7/2}+\frac {1}{176} \left (x+\sqrt {-b+x^2}\right )^{11/2}+(2 b c) \text {Subst}\left (\int \frac {\sqrt {b}-x^2}{b+x^4} \, dx,x,\sqrt {x+\sqrt {-b+x^2}}\right )-(2 b c) \text {Subst}\left (\int \frac {\sqrt {b}+x^2}{b+x^4} \, dx,x,\sqrt {x+\sqrt {-b+x^2}}\right )\\ &=-\frac {b^5}{144 \left (x+\sqrt {-b+x^2}\right )^{9/2}}-\frac {b^4}{80 \left (x+\sqrt {-b+x^2}\right )^{5/2}}+\frac {b^3}{8 \sqrt {x+\sqrt {-b+x^2}}}-\frac {b c}{\sqrt {x+\sqrt {-b+x^2}}}-\frac {1}{24} b^2 \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{3} c \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{112} b \left (x+\sqrt {-b+x^2}\right )^{7/2}+\frac {1}{176} \left (x+\sqrt {-b+x^2}\right )^{11/2}-\frac {\left (b^{3/4} c\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt {x+\sqrt {-b+x^2}}\right )}{\sqrt {2}}-\frac {\left (b^{3/4} c\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt {x+\sqrt {-b+x^2}}\right )}{\sqrt {2}}-(b c) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt {x+\sqrt {-b+x^2}}\right )-(b c) \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt {x+\sqrt {-b+x^2}}\right )\\ &=-\frac {b^5}{144 \left (x+\sqrt {-b+x^2}\right )^{9/2}}-\frac {b^4}{80 \left (x+\sqrt {-b+x^2}\right )^{5/2}}+\frac {b^3}{8 \sqrt {x+\sqrt {-b+x^2}}}-\frac {b c}{\sqrt {x+\sqrt {-b+x^2}}}-\frac {1}{24} b^2 \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{3} c \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{112} b \left (x+\sqrt {-b+x^2}\right )^{7/2}+\frac {1}{176} \left (x+\sqrt {-b+x^2}\right )^{11/2}-\frac {b^{3/4} c \log \left (\sqrt {b}+x+\sqrt {-b+x^2}-\sqrt {2} \sqrt [4]{b} \sqrt {x+\sqrt {-b+x^2}}\right )}{\sqrt {2}}+\frac {b^{3/4} c \log \left (\sqrt {b}+x+\sqrt {-b+x^2}+\sqrt {2} \sqrt [4]{b} \sqrt {x+\sqrt {-b+x^2}}\right )}{\sqrt {2}}-\left (\sqrt {2} b^{3/4} c\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {x+\sqrt {-b+x^2}}}{\sqrt [4]{b}}\right )+\left (\sqrt {2} b^{3/4} c\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {x+\sqrt {-b+x^2}}}{\sqrt [4]{b}}\right )\\ &=-\frac {b^5}{144 \left (x+\sqrt {-b+x^2}\right )^{9/2}}-\frac {b^4}{80 \left (x+\sqrt {-b+x^2}\right )^{5/2}}+\frac {b^3}{8 \sqrt {x+\sqrt {-b+x^2}}}-\frac {b c}{\sqrt {x+\sqrt {-b+x^2}}}-\frac {1}{24} b^2 \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{3} c \left (x+\sqrt {-b+x^2}\right )^{3/2}+\frac {1}{112} b \left (x+\sqrt {-b+x^2}\right )^{7/2}+\frac {1}{176} \left (x+\sqrt {-b+x^2}\right )^{11/2}+\sqrt {2} b^{3/4} c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {x+\sqrt {-b+x^2}}}{\sqrt [4]{b}}\right )-\sqrt {2} b^{3/4} c \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {x+\sqrt {-b+x^2}}}{\sqrt [4]{b}}\right )-\frac {b^{3/4} c \log \left (\sqrt {b}+x+\sqrt {-b+x^2}-\sqrt {2} \sqrt [4]{b} \sqrt {x+\sqrt {-b+x^2}}\right )}{\sqrt {2}}+\frac {b^{3/4} c \log \left (\sqrt {b}+x+\sqrt {-b+x^2}+\sqrt {2} \sqrt [4]{b} \sqrt {x+\sqrt {-b+x^2}}\right )}{\sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.59, size = 285, normalized size = 0.95 \begin {gather*} \frac {608 b^5-4620 b^3 c-6156 b^4 x^2+48510 b^2 c x^2+7470 b^3 x^4-83160 b c x^4+9510 b^2 x^6+36960 c x^6-21240 b x^8+10080 x^{10}+6 \sqrt {-b+x^2} \left (-456 b^4 x+1235 b^3 x^3+5 b^2 \left (693 c x+89 x^5\right )-20 b \left (539 c x^3+135 x^7\right )+560 \left (11 c x^5+3 x^9\right )\right )}{3465 \left (x+\sqrt {-b+x^2}\right )^{9/2}}+\sqrt {2} b^{3/4} c \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x+\sqrt {-b+x^2}}}{-\sqrt {b}+x+\sqrt {-b+x^2}}\right )+\sqrt {2} b^{3/4} c \tanh ^{-1}\left (\frac {\sqrt {b}+x+\sqrt {-b+x^2}}{\sqrt {2} \sqrt [4]{b} \sqrt {x+\sqrt {-b+x^2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[-b + x^2]*(c + x^4)*Sqrt[x + Sqrt[-b + x^2]])/x,x]

[Out]

(608*b^5 - 4620*b^3*c - 6156*b^4*x^2 + 48510*b^2*c*x^2 + 7470*b^3*x^4 - 83160*b*c*x^4 + 9510*b^2*x^6 + 36960*c
*x^6 - 21240*b*x^8 + 10080*x^10 + 6*Sqrt[-b + x^2]*(-456*b^4*x + 1235*b^3*x^3 + 5*b^2*(693*c*x + 89*x^5) - 20*
b*(539*c*x^3 + 135*x^7) + 560*(11*c*x^5 + 3*x^9)))/(3465*(x + Sqrt[-b + x^2])^(9/2)) + Sqrt[2]*b^(3/4)*c*ArcTa
n[(Sqrt[2]*b^(1/4)*Sqrt[x + Sqrt[-b + x^2]])/(-Sqrt[b] + x + Sqrt[-b + x^2])] + Sqrt[2]*b^(3/4)*c*ArcTanh[(Sqr
t[b] + x + Sqrt[-b + x^2])/(Sqrt[2]*b^(1/4)*Sqrt[x + Sqrt[-b + x^2]])]

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}-b}\, \left (x^{4}+c \right ) \sqrt {x +\sqrt {x^{2}-b}}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-b)^(1/2)*(x^4+c)*(x+(x^2-b)^(1/2))^(1/2)/x,x)

[Out]

int((x^2-b)^(1/2)*(x^4+c)*(x+(x^2-b)^(1/2))^(1/2)/x,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-b)^(1/2)*(x^4+c)*(x+(x^2-b)^(1/2))^(1/2)/x,x, algorithm="maxima")

[Out]

integrate((x^4 + c)*sqrt(x^2 - b)*sqrt(x + sqrt(x^2 - b))/x, x)

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Fricas [A]
time = 0.38, size = 269, normalized size = 0.89 \begin {gather*} -\frac {2}{3465} \, {\left (35 \, x^{5} - 19 \, b x^{3} - {\left (152 \, b^{2} - 1155 \, c\right )} x - 2 \, {\left (175 \, x^{4} - 57 \, b x^{2} - 152 \, b^{2} + 1155 \, c\right )} \sqrt {x^{2} - b}\right )} \sqrt {x + \sqrt {x^{2} - b}} + 4 \, \left (-b^{3} c^{4}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-b^{3} c^{4}\right )^{\frac {1}{4}} b^{2} c^{3} \sqrt {x + \sqrt {x^{2} - b}} - \sqrt {b^{4} c^{6} x + \sqrt {x^{2} - b} b^{4} c^{6} - \sqrt {-b^{3} c^{4}} b^{3} c^{4}} \left (-b^{3} c^{4}\right )^{\frac {1}{4}}}{b^{3} c^{4}}\right ) - \left (-b^{3} c^{4}\right )^{\frac {1}{4}} \log \left (b^{2} c^{3} \sqrt {x + \sqrt {x^{2} - b}} + \left (-b^{3} c^{4}\right )^{\frac {3}{4}}\right ) + \left (-b^{3} c^{4}\right )^{\frac {1}{4}} \log \left (b^{2} c^{3} \sqrt {x + \sqrt {x^{2} - b}} - \left (-b^{3} c^{4}\right )^{\frac {3}{4}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-b)^(1/2)*(x^4+c)*(x+(x^2-b)^(1/2))^(1/2)/x,x, algorithm="fricas")

[Out]

-2/3465*(35*x^5 - 19*b*x^3 - (152*b^2 - 1155*c)*x - 2*(175*x^4 - 57*b*x^2 - 152*b^2 + 1155*c)*sqrt(x^2 - b))*s
qrt(x + sqrt(x^2 - b)) + 4*(-b^3*c^4)^(1/4)*arctan(-((-b^3*c^4)^(1/4)*b^2*c^3*sqrt(x + sqrt(x^2 - b)) - sqrt(b
^4*c^6*x + sqrt(x^2 - b)*b^4*c^6 - sqrt(-b^3*c^4)*b^3*c^4)*(-b^3*c^4)^(1/4))/(b^3*c^4)) - (-b^3*c^4)^(1/4)*log
(b^2*c^3*sqrt(x + sqrt(x^2 - b)) + (-b^3*c^4)^(3/4)) + (-b^3*c^4)^(1/4)*log(b^2*c^3*sqrt(x + sqrt(x^2 - b)) -
(-b^3*c^4)^(3/4))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- b + x^{2}} \left (c + x^{4}\right ) \sqrt {x + \sqrt {- b + x^{2}}}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-b)**(1/2)*(x**4+c)*(x+(x**2-b)**(1/2))**(1/2)/x,x)

[Out]

Integral(sqrt(-b + x**2)*(c + x**4)*sqrt(x + sqrt(-b + x**2))/x, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-b)^(1/2)*(x^4+c)*(x+(x^2-b)^(1/2))^(1/2)/x,x, algorithm="giac")

[Out]

integrate((x^4 + c)*sqrt(x^2 - b)*sqrt(x + sqrt(x^2 - b))/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {x+\sqrt {x^2-b}}\,\left (x^4+c\right )\,\sqrt {x^2-b}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x + (x^2 - b)^(1/2))^(1/2)*(c + x^4)*(x^2 - b)^(1/2))/x,x)

[Out]

int(((x + (x^2 - b)^(1/2))^(1/2)*(c + x^4)*(x^2 - b)^(1/2))/x, x)

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